This paper introduces scour physics-informed neural network algorithms
(SPINNs), a hybrid physics-data-driven framework for bridge scour prediction
using deep learning developed based on historical scour monitoring data. SPINNs
integrate physics-based empirical equations into neural networks as
supplementary loss components. We examined three architectures: LSTM, CNN, and
NLinear. While CNN and LSTM have shown competitive real-time scour forecasting
in previous studies, NLinear with a simple architecture demonstrated the
highest accuracy and significantly lower computational cost. Despite varying
error reduction margins across different base models and bridges, SPINNs showed
promising scour prediction and generally outperformed pure data-driven models.
In some bridge cases, SPINN reduced forecasting errors by up to 50\%. In this
study, we also explored generalised models for bridge clusters by aggregating
training datasets from multiple bridges in Alaska. Bridge/site-specific SPINNs
incorporating HEC18 and time-dependent empirical equations provided more
accurate predictions than SPINNs with generalised time-dependent equations. The
three empirical equations derived from SPINN training in this study showed
reasonable accuracy in estimating maximum scour depth. These deep learning
derived empirical models can provide more accurate and reliable scour
predictions than traditional HEC-18, particularly in scenarios lacking
site-specific scour data. Comparing the HEC-18 model with both SPINNs and pure
deep learning models highlights a substantial improvement in scour prediction
accuracy, indicating a promising future for these hybrid machine learning
methodologies for bridge scour design and maintenance.

该研究介绍了一种基于历史冲刷监测数据开发的基于物理和数据驱动的混合框架，称为 SPINNs（Scour Physics-Informed Neural Network Algorithms），用于桥梁冲刷预测的深度学习算法。通过将物理基于经验的方程式作为辅助损失组件，SPINNs 将这些方程式整合到神经网络中。本研究通过考察三种架构：LSTM、CNN 和 NLinear，发现简单架构的 NLinear 具有最高的准确性和较低的计算成本。相比纯数据驱动模型，SPINNs 在冲刷预测方面表现更加优越。该研究还探索了将多座桥梁的训练数据集聚合为桥梁群的泛化模型，并发现与时间相关的经验方程式可以提供更准确的预测结果。使用 SPINNs 训练得出的三个经验方程式在估计最大冲刷深度方面具有合理的准确度。相比传统的 HEC-18 模型，SPINNs 和纯深度学习模型在冲刷预测准确性方面均有显著改善，为桥梁冲刷设计和维护提供了有前景的混合机器学习方法。

引入物理可知的深度学习框架用于桥梁冲刷预测

Introducing a Physics-informed Deep Learning Framework for Bridge Scour  Prediction

In this study, we introduce a method based on Separable Physics-Informed
Neural Networks (SPINNs) for effectively solving the BGK model of the Boltzmann
equation. While the mesh-free nature of PINNs offers significant advantages in
handling high-dimensional partial differential equations (PDEs), challenges
arise when applying quadrature rules for accurate integral evaluation in the
BGK operator, which can compromise the mesh-free benefit and increase
computational costs. To address this, we leverage the canonical polyadic
decomposition structure of SPINNs and the linear nature of moment calculation,
achieving a substantial reduction in computational expense for quadrature rule
application. The multi-scale nature of the particle density function poses
difficulties in precisely approximating macroscopic moments using neural
networks. To improve SPINN training, we introduce the integration of Gaussian
functions into SPINNs, coupled with a relative loss approach. This modification
enables SPINNs to decay as rapidly as Maxwellian distributions, thereby
enhancing the accuracy of macroscopic moment approximations. The relative loss
design further ensures that both large and small-scale features are effectively
captured by the SPINNs. The efficacy of our approach is demonstrated through a
series of five numerical experiments, including the solution to a challenging
3D Riemann problem. These results highlight the potential of our novel method
in efficiently and accurately addressing complex challenges in computational
physics.

通过引入基于 SPINNs 的方法解决 Boltzmann 方程的 BGK 模型，以改善积分计算和近似粒子密度函数的困难，提高了计算物理中复杂挑战的处理效率和准确性。